Word for a person at a company who is in charge of my data
Solution 1:
From Oxford Advanced Learner's:
liaison - communication or cooperation that facilitates a close working relationship between people or organizations
My liaison at the insurance company is in charge of my case.
Perhaps, for a less formal-sounding option, the word agent is colloquially used.
From Merriam-Webster:
agent - a person who does business for another person : a person who acts on behalf of another
One thing you might find about English is that it hasn't quite caught up to the "Digital Age". One user points this out fairly well. There just aren't many single-words that encapsulate exactly what's needed. Terms like data owner and data custodian exist, but, as @JohnLawler points out, English just isn't quite there yet.
Solution 2:
With many service providers, I've run into account manager, as the person in charge of my account at their company, and thus my contact with the company, who's familiar with my needs and history with the company.
Solution 3:
In the "service firm" sector, another possibility related to liaison is "account rep" or "account representative", the person at a company who deals with matters concerning a subset of customers/clients/accounts to which they have been assigned.
In an insurance company, it could be a "claims adjuster" -- the person who is handling a claim you submitted.
Solution 4:
It sounds to me like it should be (insurance)
case manager
or (insurance)